attempt 1: 10 attempts remaining. the number of hats made by a manufacturer t days after the start of the…

attempt 1: 10 attempts remaining. the number of hats made by a manufacturer t days after the start of the month is modeled by n(t)=20t² + 6t, and the cost per hat, in dollars, is modeled by c(t)=ln(228t). the total cost of the hats made is v(t)=c(t)·n(t). how fast is the total cost of the hats changing on the 12th day? (round your answer to the nearest cent.)

attempt 1: 10 attempts remaining. the number of hats made by a manufacturer t days after the start of the month is modeled by n(t)=20t² + 6t, and the cost per hat, in dollars, is modeled by c(t)=ln(228t). the total cost of the hats made is v(t)=c(t)·n(t). how fast is the total cost of the hats changing on the 12th day? (round your answer to the nearest cent.)

Answer

Explanation:

Step1: Find the product - rule formula

The product - rule states that if (v(t)=c(t)\cdot n(t)), then (v^\prime(t)=c^\prime(t)n(t)+c(t)n^\prime(t)). First, find (n^\prime(t)) and (c^\prime(t)). For (n(t) = 20t^{2}+6t), using the power - rule ((x^n)^\prime=nx^{n - 1}), we have (n^\prime(t)=\frac{d}{dt}(20t^{2}+6t)=40t + 6). For (c(t)=\ln(228t)), using the chain - rule ((\ln(u))^\prime=\frac{u^\prime}{u}), where (u = 228t) and (u^\prime=228), so (c^\prime(t)=\frac{228}{228t}=\frac{1}{t}).

Step2: Substitute (t = 12) into (n(t)), (n^\prime(t)), (c(t)) and (c^\prime(t))

  • Calculate (n(12)): (n(12)=20\times12^{2}+6\times12=20\times144 + 72=2880+72 = 2952).
  • Calculate (n^\prime(12)): (n^\prime(12)=40\times12 + 6=480+6=486).
  • Calculate (c(12)): (c(12)=\ln(228\times12)=\ln(2736)\approx7.91).
  • Calculate (c^\prime(12)): (c^\prime(12)=\frac{1}{12}).

Step3: Calculate (v^\prime(12))

Using the product - rule (v^\prime(12)=c^\prime(12)n(12)+c(12)n^\prime(12)) (v^\prime(12)=\frac{1}{12}\times2952+7.91\times486) (v^\prime(12)=246+7.91\times486) (v^\prime(12)=246 + 3844.26) (v^\prime(12)=4090.26)

Answer:

(4090.26)